What is Differentiation: Definition and 1000 Discussions

The cluster of differentiation (also known as cluster of designation or classification determinant and often abbreviated as CD) is a protocol used for the identification and investigation of cell surface molecules providing targets for immunophenotyping of cells. In terms of physiology, CD molecules can act in numerous ways, often acting as receptors or ligands important to the cell. A signal cascade is usually initiated, altering the behavior of the cell (see cell signaling). Some CD proteins do not play a role in cell signaling, but have other functions, such as cell adhesion. CD for humans is numbered up to 371 (as of 21 April 2016).

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  1. A

    Optimization using differentiation

    Homework Statement The owner of a luxury motor yacht that sails among the 4000 Greek islands charges $600 per person per day if exactly 20 people sign up for the cruise. However, if more than 20 people sign up (up to the maximum capacity of 90) for the cruise, then each fare is reduced by $4...
  2. A

    Optimization using differentiation

    Homework Statement http://i53.tinypic.com/1zu5ty.jpg The Attempt at a Solution well so far, all i got is 3x + y = 3000; also y = 1000/x ==>> 3x+ (1000/x) = 3000 i don't know what the area should be though... would it be A=x2y If i am right in that, would i do this after i...
  3. S

    Hard logarithmic/implicit differentiation question.

    QUESTION HAS BEEN SOLVED ! :) THANKS MARK! Homework Statement Shark Inc. has determined that demand for its newest netbook model is given by https://webwork.elearning.ubc.ca/webwork2_files/tmp/equations/80/fa9842dd3479ee1baaab655e02a1c41.png, where q is the number of netbooks Shark can sell...
  4. S

    Anti differentiation of f(x)f'(x)

    Homework Statement \int(x3+x)\sqrt{x4+2x2}dx Homework Equations \int[f(x)r]f'(x)dx=(([f(x)]r+1)/(r+1))+C The Attempt at a Solution The x3+x part is close to the derivative of the other part that I'm supposed to be anti differentiating, so should I just introduce a 4/4 into the...
  5. A

    Why Is There Only One Stationary Point for the Volume of This Box?

    Homework Statement The volume of a box is given by V=4x^3 -66x^2 +216x. Explain why there is only one value of x for which V is stationary? Homework Equations The Attempt at a Solution dV/dx = 12x^2 -132x +216 at a stationary point x^2-11x+18=0 so x=2 or 9
  6. A

    How Do You Differentiate tan-1(xy) in Implicit Differentiation?

    Homework Statement Find y' given tan-1(xy) = 1 + x2y Homework Equations The Attempt at a Solution I know I'll have to start with implicit differentiation, and I can differentiate the RHS to be: 0 + 2xy + x2(dy/dx) And I know tan-1 x = 1/(1+x2), but I don't know how to...
  7. A

    Does this differentiation trick work?

    Hey all, I came up with a trick for fast differentiation, and it's worked every time I've used it, but I was hoping you guys could possibly formally prove that it works, or at least show me a case where it doesn't? It goes like this: (Sorry I can't figure out Latex) When you have an...
  8. S

    Equation Manipulation with Differentiation or Integration?

    Solve the equation below to gain an expression for θ(t), where v, β and r₀ are constants and the only variables are time(t) and theta(θ): vt=r₀θ+βθ²/2 The exercise is based upon the spiral data track of a compact disc and either side of the equation is equal to the displacement. I’ve...
  9. Telemachus

    Second derivatives using implicit differentiation

    Hi there. Well, I wanted to know how to find the second derivatives of a function using implicit differentiation. Is it possible? I think it is. I think I must use the chain rule somehow, but I don't know how... I'm in multivariable calculus since the function I'm going to use could be seen as a...
  10. N

    Understanding Partial Differentiation in Classical Mechanics

    Hello, I got confused in my Classical Mechanics class (on a mathematical issue). So let L denote a function dependent on x and its derivative explicitly, such that its image is L(x,x*) (NOTE: I'm using * as the overdot-Leibniz notation for the derivative) and x is a function of t. To make...
  11. A

    Implicit differentiation help again

    implicit differentiation help :) again! Homework Statement Use implicit differentiation 1) x/(y-x2)=1 and 2) (y2-1)3=x2-y The Attempt at a Solution 1) x/(y-x2)=1 => [(y-x2)(1)-(x)((1*dy/dx)-2x)]/[(y-x)2]2=0 => y-x2-x(dy/dx)+2x2=[(y-x)2]2 I think I'm going to stop here because I'm pretty...
  12. A

    Horizontal tangents via implicit differentiation

    Homework Statement Find the points (if any) of of horizontal tangent lines on : x2 + xy + y2 = 6 Homework Equations n/a The Attempt at a Solution So far I've concluded that I must find the points at which dy/dx = 0. I've solved for dy/dx and arrived at dy/dx = (-2x-y)/(x+2y)...
  13. C

    Complicated implicit differentiation

    Homework Statement If (sqrt x) + (sqrt y) = 11 and f(9)=64 ---> find f '(9) by implicit differentiation The Attempt at a Solution I keep getting lost in my work here... first, taking derivative of both sides: d/dx ((sqrt x) + (sqrt y)) = d/dx (11)...
  14. C

    Differentiation of exponents x

    Homework Statement If f(x) = 2sin(x) + (4x)^x find ---> f ' (3) Homework Equations d/dx (e^u) = e^u du/dx The Attempt at a Solution Here is my attempt: f(x) = 2sin(x) + e^(ln(4x)^x) = 2sin(x) + e^(xln4x) = d/dx (2sin(x)) + d/dx (e^(xln4x)) = 2cos(x)...
  15. S

    What Is the Correct Derivative of the Function p(t)?

    Homework Statement Hi can anyone tell me what is the derivative of this function. Homework Equations p(t) = u(X(t), t) The derivative of p with respect to t The Attempt at a Solution dp/dt = du/dX * dX/dt + 1 ?? or is it going to be du/dX * dX/dt
  16. D

    Implicit Differentiation and Complex numbers concept question

    I just started learning Implicit Differentiation and came across an issue. I took the derivative of the circle function: y2 + x2 = 1 y' = -x / y This all made sense until I solved the circle function for y, which gives: y = \pm\sqrt{1 - x^2} For any x > 1, it's going to be complex. So, does...
  17. H

    Implicit Differentiation Tangent lines

    Homework Statement Where does the graph of 25x^2 + 16y^2 + 200x - 160y + 400 = 0 have a horizontal tangent line. Homework Equations dy/dx dx/dy or something not sure. The Attempt at a Solution Well I know that a horizontal tangent line would mean the slope is zero...but what...
  18. M

    Implicit partial differentiation

    Homework Statement Find \partial x / \partial z at the point (1, -1, -3) if the equation xz + y \ln x - x^2 + 4 = 0 defines x as a function of the two independent variables y and z and the partial derivative exists. Homework Equations The Attempt at a Solution x + y/x \partial x...
  19. D

    How do you differentiate logarithms with different bases and exponents?

    How do you differentiate : 2^n/2? You can't you just use the power rule? The correct answer is 2^n/2 (In^2) 1/2 lg^2 x n Where lg is log base 2. The correct answer is 2lgn 1/n^2 . 1/n Why is this so? Isn't lg^2 = 1? And differentiate n and we get 1? Sorry if I sound...
  20. A

    How Do You Isolate dy/dx in Implicit Differentiation?

    implicit differentiation help :) Homework Statement Find dx/dy by implicit differentiation (x2+ y)2+ x2+ xy2= 100Homework Equations The Attempt at a Solution I'm trying to use the chain rule to solve it... i got The derivative ofb(x2+ y)2+ x2+ xy2= 100, with respect to x, is 2(x2+...
  21. A

    Implicit Differentiation: Solving for dx/dy in an Example Problem

    Let me first say i just learned implicit differentiation (earlier today) and i am also new in general to derivatives. I am finding implicit differentiation difficult but i want to understand it before we go over it in class. Homework Statement This is a example in my book. I have been trying...
  22. P

    Optimizing Power Transmission with Differentiation: Solving for Maximum Velocity

    Homework Statement (a) A belt of mass m per unit length is wound partly around a pulley. The power P(v) transmitted is given by P=Tv-mv3 where T is the tension and v is velocity of the belt. Use differentiation to show that the maximum transmission occurs when v=sqrt(T/3m). Homework...
  23. P

    Implicit differentiation - Help

    Homework Statement find y' given x3+y2+x+y=2 Homework Equations The Attempt at a Solution dy/dx=3x2+2y(y')+1+y' I know that's wrong because my 89 gives it as y'=(-3x2+1)/(2y+1), I don't know how to get there though and what happened to the =2. Thanks
  24. S

    Superior implicit differentiation, prove answer.

    Homework Statement 2XY = Y^2 prove that y''2 = \frac{y^{2}-2xy}{(y-x)^3} EDIT: Sorry, don't know how to insert a space in Latex. Homework Equations The Attempt at a Solution 2y+2x \frac{dy}{dx} = 2y \frac{dy}{dx} \frac{dy}{dx}(2y-2x) = 2y \frac{dy}{dx}= \frac{y}{y-x}...
  25. M

    Differentiation and finding tangent

    Homework Statement Find the equation of the tangent to the curve y = x2(x + 1)4 at the point P(1,16) Homework Equations The Attempt at a Solution dy/dx x2(x + 1)4 = (x + 1)3((x + 1)2x + 4x2) = (x + 1)3(6x2 + 2x) = (x + 1)3(2x)(3x + 1) Subst. 1 into find grad. (1 +...
  26. J

    Implicit Differentiation Shortcut

    Homework Statement differentiate: yx = y / sqrt(x2 + y2) Homework Equations The Attempt at a Solution I solved this problem by taking the ln of both sides and then solving. It seems from the context of the problem set that this was supposed to be easier than that. Am I...
  27. T

    Sketching A Graph Using Differentiation

    Homework Statement Homework Equations The Attempt at a Solution I don't understand how they get from step 4 to step 5. Wouldn't you factor a cos x out of the brackets then have (\cos x)(1-\frac{1}{6}) to the left of the brackets. Then you can multiply the (1-\frac{1}{6}) inside...
  28. T

    Mastering Differentiation Calculus: Solving Homework Equations with Ease

    Homework Statement Differentiate Homework Equations see 1. The Attempt at a Solution btw the x in the second equation is meant to be an r how much did i fail? :(
  29. L

    Implicit differentiation, whats going wrong?

    Implicit differentiation, what's going wrong!? Hey people can someone point out to me please where I'm going wrong with part B of this question, can't get it to look the answer in the book, d2y/dx2 = -3(x^6/y^7)- 3(x^2/y^3). My answer is listed below under part B section, but i can't...
  30. E

    Understanding Water Flow in a Leaky Container Using Differentiation

    Homework Statement dV/dT = 0.15(300-V) V is the volume of water in a container at time T. The rate of water flowing into the container is constant at 45 units^3/s. The rate of water leaking is proportional to the volume of water currently in the container. Find the volume of water in the...
  31. K

    Help for quickest and clearest route for differentiation question

    Homework Statement Find the coordinates of the points on these curves at which the gradient has the given values. y=1/3x^(3/2)-x^(1/2), gradient=3/4 The Attempt at a Solution Basically the problem I have isn't really finding the answer because... x^(1/2)-x^-(1/2)=6/4[/b] and...
  32. V

    How Do I Correctly Simplify the Derivative of Arctan(2x-3)?

    Homework Statement \frac{d}{dx} [ arctan (2x-3) ] The Attempt at a Solution The general solution for d/dx arctan(x) = \frac{du/dx}{1+x^2} so \frac{du/dx}{1+(2x-3)^2} du/dx = 2 \frac{2}{1+(2x-3)^2} \frac{2}{1+2x^2-12x+9} \frac{2}{10+2x^2-12x} now this is where my question...
  33. A

    Differentiation and Lebesgue integration

    Homework Statement Suppose g(x) = \int_0^x f(t) dt, where f is Lebesgue integrable on \mathbb R. Give an \epsilon - \delta proof that g'(y) = f(y) if y\in (0,\infty) is a point of continuity of f. Homework Equations The Attempt at a Solution I know I need to show that f(y) =...
  34. E

    What is the total differential for a function with two variables?

    Homework Statement a=(t^2 + 1)i - j + tk Find da/dt I multiplied out the brackets so a= it^2 + i - j + tk then got as far as da/dt= 2it + 0 + 0 + ? a) is this right so far? b) how do i integrate tk with respect to t? Any help would be greatly appreciated.
  35. T

    Why is the derivative of cot x equal to -cosec^2 x instead of -1/tan^2?

    Can someone show me why the derivative of k(x) = cot x Is k'(x) = -cosec^2 x And not k'(x) = -1/tan^2 Thanks!
  36. C

    MATLAB Symbolic Differentiation in Matlab

    Hi, quick question. I have an inline function I would like to differentiate symbolically in Matlab, however when I do it I get the following: say x = inline('cos(t.^2)','t') ; xp = inline(char(diff(x(t),sym('t'))),'t') ; xp(t) becomes -sin(t^2)*t however I really...
  37. G

    Differentiation of a Piecewise Function

    Homework Statement I have a graph of velocity vs time and was asked "Describe how the velocity changes over the time interval of the problem" how do I do this all I have is the graph that is not defined by a function... well I can make my own strangely all it is is a bunch of linear lines...
  38. P

    Implicit differentiation - Help

    Homework Statement Find d2y/dx2 for cos y + x2 =12 The first derivative I think is correct but the second I am unsure of, this is Q5 of 10 in an assignment and every other question has been far cleaner and easier making me think that I've missed something. We don't have to use implicit...
  39. B

    Implicit Differentiation; Tangent Line

    Homework Statement Find the equation of a tangent line at the curve at point (-3√3, 1) x^(1/3) + y^(1/3) = 4Homework Equations Point-slope: y-1=m(x-1) The Attempt at a Solution I took the derivative of that equation and resulted in -y^(2/3)/x^(2/3) When I tried plugging in x and y to...
  40. B

    Another differentiation problem

    Homework Statement If z² = x² + y² and dx/dt = 2 and dy/dt = 3 Find dz/dt when x=5 and y=12 Homework Equations I used The Attempt at a Solution But I resulted in: 2z dz/dt = 2x(dx/dt) + 2y(dy/dt) Plugged in x,y,dx/dt, and dy/dt to get: 2z (dz/dt) = 2(5)(2) + 2(12)(3) dz/dt = 80/2z...
  41. B

    What is the Derivative of a Cubic Function with a Given Value of x?

    Homework Statement If y = x³ + 2x and dx/dt = 5, find dy/dt when x = 2. The Attempt at a Solution I'm not really sure how to solve this problem at all. It's the first I've seen like this so I don't know what approach I should take.
  42. B

    Differentiation of log function

    Homework Statement Find an equation of the tangent line to the curve at the given point. y = ln(xe^(x²)) P=(1, 1) The Attempt at a Solution I know how to plug in the numbers into the Point-Slope formula once I find the derivative of the function, or slope, I'm just not sure...
  43. P

    Differentiation from first principles- - cant do at all

    Homework Statement see attachment- have to do this because i can't figure out how to do the notation in this part sorry... I have no idea where to go with this and probably need quite a bit of help with it--- thanks. Homework Equations The Attempt at a Solution
  44. P

    Differentiation - cant remember

    Homework Statement for the question and attempt see the attachment. I've nearly done this but from my ti-89 I know that i am missing a 2x from the top line and i can't find where its gone. been a while since I've done this and i think that its just a simple mistake... ps it does want the...
  45. S

    Implicit differentiation and optimization

    Homework Statement A conical tent must contain 40\pi ft^{3}. Compute the height and radius of the tent with minimal total surface area. (Include the floor material.) Homework Equations 1. \frac{\pi r^{2} h}{3} = 40\pi 2. \pi r \sqrt{r^{2} + h^{2}} + \pi r^{2} = S 3. \frac {dr}{dh} =...
  46. C

    Finding dy/dx of y=1/(x+y) using implicit differentiation: Step-by-step guide

    can someone show me step by step how to find dy/dx of y=1/(x+y) using implicit differentiation?
  47. B

    Differentiation find dy/dx if y=x^y

    I need to check my results on this integration find dy/dx if y=x^y I find: dy/dx= y^2/[x(1-yln(x))]
  48. N

    Is Re(f) and Im(f) always differentiable for complex functions?

    Hello. I was wondering if you have a complex function f: A \subset C \to C everywhere differentiable on A, can you say anything about the existence of \frac{d Re(f)}{d z} or \frac{d Im(f)}{d z}? I was thinking that it's easy to proof that if f : A \subset R \to C, then Re(f) and Im(f) are...
  49. D

    Implicit differentiation and the product rule

    An example of implicit differentation in Stewart, 6th ed, p 883, is given as follows: x^3 + y^3 + z^3 + 6xyz = 1 Differentiating to find dz/dx, 3x^2 + 3z^2(dz/dx) + 6yz + 6xy(dz/dx) = 0 where the product rule was used to differentiate 6xyz with respect to x. Why isn't the...
  50. K

    Covariant differentiation commutes with contraction?

    Homework Statement I've been reading a textbook on tensor analysis for a while. The book uses the conclusion of "covariant differentiation commutes with contraction" directly and I searched around and found most people just use the conclusion without proof.Homework Equations For example...
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